Apparatus and method for determining the correlation coefficient of two graphical traces



Aug. 4, 1970 ,E. F. KILPATRICK ET AL 3,522,655

APPARATUS AND METHOD FOR DETERMINING THE CORRELATION COEFFICIENT OF TWOGRAPHICAL TRACES Filed 001; 2, 1967 JOHN BEMROSE, deceased by ELSIEHILDEGARD BEMROSE, Administrmrix ELLEN F KILPATRICK ATTORNEYS.

United States Patent 3,522,655 APPARATUS AND METHOD FOR DETERMINING THECORRELATION COEFFICIENT OF TWO GRAPHICAL TRACES Ellen F. Kilpatrick,Tulsa, Okla., and John Bemrose, deceased, late of Tulsa, Okla., by ElsieHildegard Bemrose, administratrix, Tulsa, Okla., assignors to Sinclair'Research, Inc., New York, N.Y., a corporation of Delaware Filed Oct. 2,1967, Ser. No. 672,397 Int. Cl. G06g 3/00 US Cl. 33--1 7 Claims ABSTRACTOF THE DISCLOSURE An apparatus for and a method of determining thecorrelation coefficient p (x, y) of two graphical traces, x and y. Ahorizontal reference line is located through each trace, for example atits mean value. One trace is mounted horizontally upon a slider movazlewithin the body of the correlation apparatus, and the other trace ismounted horizontally directly on the body of the instrument, above theslider. The slider is moved to vertically align the traces at whatappears to be good correlation. A plurality of equispaced referencepoints are located along the traces. At each reference point the twotraces are deemed to correlate if they both lie on the same side oftheir reference line. If they lie on opposite sides, they are deemed notto correlate. The correlation coefficient p (x, y), is then given by 12)l 1Nz)] where N is the number of reference points at which the tracescorrelate and N is the number of reference points at which the traces donot correlate.

c =sin DESCRIPTION OF THE INVENTION Numerous analytical testing andevaluation procedures include the development and interpretation of agraphical trace. By way of example,electrocardiograms, X-ray diffractionanalyses, and the processing of seismic data frequently include themaking and analysis of graphical traces. Such an analysis often includesdetermination of correlation between the graphical trace and a known orreference pattern. Such analysis permits the determination of the degreeof correlation or the degree of deviation of the test or sample tracewith the reference pattern. Then the degree of correlation of severaltraces can be compared by comparing their correlation coefficients. Inlike manner, a given test trace can be compared with several referencepatterns to determine the reference pattern to which it bears thehighest degree of correlation. In such instances the determination of acorrelation coefficient permits the necessary determinations to be made.

Numerous mathematical formulae exist for the determination of thecorrelation coefficient between two variables. In general, theseformulae are extremely complex, and a great expenditure of time andenergy is required to develop a correlation coeflicient from them.

-Thus, the correlation coefficient p (x, y) between two probabilityvariables x and y is given by:

L xyPo, mm

may -w where 0x and .ry are the variances of x and y respec- The presentinvention is an apparatus and a method for determining the correlationcoeflicient of two graphical traces from the observation of thefrequency with which the two traces deviate in like direction from theirrespective mean values. The apparatus and method permit a rapid andsimple determination of the correlation coeflicient p(x, y) of the twographical traces.

A complete appreciation of the aspects and advantages of the presentinvention can be obtained from the following detailed description andclaims, particularly when read in conjunction with the accompanyingdrawings in which like parts bear like reference numerals.

In the drawings:

FIG. 1 is a plan view of the correlation measuring apparatus of thepresent invention; and

FIG. 2 is a sectional view taken along line 2-2 of FIG. 1.

The correlation measuring apparatus of the present invention isgenerally indicated by reference numeral 10 in FIGS. 1 and 2. Long thinrectangular body 12 has a longitudinal recess 14 in its upper surface.Slider 16 fits within recess 14 and is held in place by flanges 18 and20 which mate into grooves in body 12. Thus slider 16 can movelongitudinally with respect to body 12.

The sample graphical trace to be correlated is drawn upon paper strip22. Thus, in FIG. 1, a function x is represented by graphical trace 24,drawn on paper strip 22. Strip 22 is mounted upon slider 16 and is heldin place by suitable means (not shown) such as masking tape, clips, orsoft paste. In FIG. 1, the strip 22 containing the sample graphicaltrace 24 is depicted as having a length considerably less than thelength of slider 16. However, strip 22 might be of any length up to thelength of slider 16, and in the event the graphical trace to becorrelated is of a length greater than the length of slider 16, then thestrip 22 can be cut into suitable sections to be individuallycorrelated. Trace 24 might be the output of an oscillograph, by way ofexample.

Paper strip 26 containing a reference graphical trace 28 is mounted uponthe surface of body 12 above the slider 16. Reference graphical trace 28might represent a function y obtained as data from a previous test as astandard reference trace, or as any other suitable source of referencedata. Paper strip 26 is likewise held to body 12 by any suitable meanssuch as masking tape, clips, or soft paste. If a standard referencegraphical trace is utilized as graphical trace 28, then it might bepermanently marked on the surface of body 12, rather than attached to itby means of paper strip 26.

The length of body 12 is divided into a large number of equally spaceddivisions by vertical lines 30 which are spaced 2. short distance apart,for example, a spacing of inch apart. Thus each vertical line 30establishes a reference point along the length of body '12. The verticallines 30 are partially covered by strips 22 and 26, but if strips 22 and26 are of a clear material such as photographic film, then lines 30 areentirely visible. This, is however optional.

A movable member or cursor 32 is mounted upon body 12 and is held inplace by flanges 34 and 36 which mate into grooves on the upper andlower edges of body '12. Cursor 32 is made of a transparent materialsuch as a clear plastic so that traces 24 and 28 may be clearly seenthrough it. A vertical index line 38 is marked on the surface of cursor32 to permit vertical correlation of points on curves 24 and 28.

When the correlation between curves 24 and 28 is to be determined, ahorizontal reference line is drawn through each curve. Preferably thisreference line passes through the average or mean value of the curve.Thus reference line 40 is drawn through test graphical trace 24 atsubstantially the mean value of the graphical trace. Similarlyhorizontal reference line 42 is drawn through reference graphical trace28 at substantially the mean value of that graphical trace. Paper strip22 is then mounted upon the surface or slider 16, and the position ofslider 16 is adjusted to a position of apparent correlation of graphicaltraces 24 and 28. Of course it might not be possible to determine thebest correlation position, and in such instances the correlationcoefficient can be determined by the present method at two or morepositions of good correlation, and then the best correlation positionwill be known from these results. Additionally, if desired, the testgraphical trace can be divided into segments, and the correlation ofeach segment determined, instead of determining the correlation of thetest trace as a whole. The segments into which the test trace is dividedcould overlap, if desired.

In the representative example depicted in FIG. 1, slider 16 is adjustedto a position at which there is a high degree of correlation betweengraphical traces 24 and 28. Cursor 32 is then moved to a position inwhich index line 38 is aligned with the reference point established bythe vertical line 30 closest to the left edge of graphical trace 24. Anobservation is then made of the positions of traces 24 and 28 withrespect to their reference lines 40 and 42. Either both traces 24 and 28lie above their respective reference lines at the point at which indexline 38 crosses them, or they both lie below their respective referencelines, or one lies above its reference line and the other below itsreference line. Cursor 32 is then moved to align index line 38 with thenext reference point 30 along the graphical trace 24. Anotherobservation is made of the positions of graphical traces 24 and 28 withrespect to their reference lines 40 and 42. This process is repeateduntil observations of the positions of the graphical traces have beenmade at each reference point 30 along the length of graphical trace 24.

If at a given reference point 30 both graphical trace 24 and graphicaltrace 28 are displaced above their respective reference lines, then theyare deemed to correlate at that location. Likewise, if both graphicaltraces 24 and 28 are displaced below their respective reference lines 40and 42 they are deemed to correlate at that position. However, if onegraphical trace is displaced above its reference line and the otherbelow its reference line, then the two graphical traces 24 and 28 aredeemed not to correlate at that particular reference point. A count ismade of the number of reference points 30 at which graphical traces 24and 28 correlate. Likewise a count is made of the number of referencepoints at which graphical traces 24 and 28 do not correlate. If Nrepresents the number of reference points 30 at which the graphicaltraces 24 and 28 correlate by either both lying above their respectivereference line or both lying below their respective line, and Nrepresents the number of reference points 30 at which graphical traces24 and 28 do not correlate because one of the graphical traces liesabove its reference line and the other graphical trace lies below itsreference line, then the correlation number R is given y N N2 C Fromthis value of R, the correlation coefiicient of the two probabilityvariables x and y is found from the relation p y)=s n Rt a w] A morecomplete derivation of this relation is found in the paper A SimpleMethod for Calculating the Correlation Coeflicients by Yoshibumi Tomoda,Journal of Physics of the Earth, vol. 4, No. 2 (1956).

When the two traces are in perfect correlation, they have nodisplacements in opposite phase, and therefore N is zero. In such a casethe correlation coefficient p (x, y) is equal to +1.0. At the oppositeextreme, if the traces are in perfect anti-correlation, all of thedisplacements are in opposite phase, and therefore N is 0. In such acase the correlation coefficient equals --1.0. Thus the correlationcoefiicient varies from +1.0 to 1.0 depending upon the degree ofcorrelation of the two graphical traces, the same as the normalizedcorrelation coefiicient determined by complex mathematical methods.

The degree of significance of the correlation coefficient depends uponthe number of reference points 30 over which the correlation is made.Thus, the more closely spaced the reference points 30 are, the greaterthe significance of the correlation coeflicient. Likewise, a moresignificant correlation coetficient can be obtained for a long tracethan for a short one.

Although the present invention has been described with respect to apreferred embodiment and method, obviously numerous modifications andvariations could be resorted to and still be within the scope of theinvention which is defined more particularly by the claims.

What is claimed is:

1. A method of determining a correlation coefiicient p(x, y) of twographical traces x and y comprising the steps of:

(a) locating a horizontal reference line through each of said traces;

(b) placing said traces in vertical juxtaposition;

(c) locating a plurality of equispaced reference points along saidtraces;

(d) determining the number N of said reference points at which the twotraces are displaced in the same direction from their respectivereference lines;

(e) determining the number N of said reference points at which the twotraces are displaced in opposite directions from their reference lines;and

(f) calculating the correlation coeflicient of said two traces inaccordance with the relation 2. The method of claim 1 in which each saidhorizontal reference line is located through the mean value of itsrespective trace.

3. The method of claim 1 further including the step of adjusting thevertical alignment of said traces to a position of apparent correlationprior to determining N and 4. Apparatus for determining correlationbetween a reference graphical trace and a sample graphical trace, eachtrace having a reference line, by locating a plurality of equispacedreference points along said graphical traces to permit determination ofthe correlation of said graphical traces relative to their respectivereference lines at each said reference point, whereby a correlationcoefficient can be calculated from the results of said determinations,said apparatus comprising:

(a) a reference graphical trace;

(b) holding means for holding said reference graphical trace and saidsample graphical trace in juxtaposition comprising a base member forholding one of said traces and a slider member mounted on said basemember in a slideable manner for holding the other of said tracesadjacent to said one trace, whereby the alignment of said traces can beadjusted to a position of apparent correlation; and

(c) locating means comprising a plurality of equispaced reference pointson said base member.

5. Apparatus as claimed in claim 4 in which said base member comprisingan elongated body having a recess formed in one surface and throughoutthe length thereof, said recess having grooves formed on the sidesthereof, and in which said slider member comprises an elongated bodyslidably mounted within said recess and having flanges which mate withsaid grooves to hold said slider member within said recess.

6. Apparatus as claimed in claim 4 further comprising a cursor slidablymounted on said base member and having an index line overlaying bothsaid traces.

7. Apparatus as claimed in claim 4 in which said sample graphical tracecomprises a seismic data trace.

References Cited UNITED STATES PATENTS 2,207,375 7/1940 Friedell 23564.72,348,411 5/ 1944 Petty.

2,359,837 10/ 1944 Freedlander 235--70 2,455,522 12/ 1948 Ringler -i235-70 X 2,599,861 6/1952 Poole.

2,977,680 4/1961 Halverson.

10 WILLIAM D. MARTIN, JR., Primary Examiner U.S. Cl. X.R. 33107, 235-70

